Consider the model of consumption under uncertainty from the chapter 2 notes. Let the per period utility function be quadratic: u(c) =c- a2 c2. The household discounts the future at rate p. Future income, is uncertain, but evolves according to: y' = a+By+', where a, B > 0, and ' is a random variable with mean zero. Assume that parameter values are such that the present value of the consumer's lifetime income is positive for all possible realizations of . The household has rational expectations. Suppose that consumption is taxed at a constant proportional rate of t in both periods. You can think of this as a sales tax on all purchases of consumption goods (like the GST in Canada). a) Formally write down the household's optimization problem. Write down the Lagrangian function. Interpret the Lagrange multiplier. b) Derive and interpret the first order conditions. c) Assume thatp = r. Solve for the optimal value of c. Interpret your expression.